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Chapter 2: Problem 20

Differentiate implicitly to find \(d y / d x\). $$y^{3}=x^{5}$$

Short Answer

Expert verified
\( \frac{dy}{dx} = \frac{5x^4}{3y^2} \)

Step by step solution

01

Differentiate both sides with respect to x

Start by differentiating both sides of the equation with respect to x. Remember to use the chain rule when differentiating y, since y is a function of x: \[ \frac{d}{dx} (y^{3}) = \frac{d}{dx} (x^{5}) \]
02

Apply the chain rule

Use the chain rule to differentiate the left side of the equation. The chain rule states that \( \frac{d}{dx} (y^3) = 3y^2 \frac{dy}{dx} \): \[ 3y^2 \frac{dy}{dx} = 5x^4 \]
03

Solve for \( \frac{dy}{dx}

To isolate \( \frac{dy}{dx} \), divide both sides of the equation by \( 3y^2 \):\[ \frac{dy}{dx} = \frac{5x^4}{3y^2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chain Rule
The chain rule is a fundamental technique in calculus for differentiating composite functions. When you have a function composed of two or more functions, the chain rule helps you differentiate these 'nested' functions. In our exercise, we deal with the function \(y^3\), where \(y\) is itself a function of \(x\). The chain rule states that to differentiate \(y^3\) with respect to \(x\), you first differentiate \(y^3\) with respect to \(y\) and then multiply by the derivative of \(y\) with respect to \(x\). Here鈥檚 the mathematical expression:
  • Differentiate \( y^3 \) with respect to \( y \): \( \frac{d}{dy}(y^3) = 3y^2 \)
  • Then, multiply by \( \frac{dy}{dx} \) which is the derivative of \( y \) with respect to \( x \): \( 3y^2 \frac{dy}{dx} \)
This method is essential when dealing with implicit functions, where \(y\) cannot be easily isolated.
Differentiation
Differentiation is the process of finding the derivative of a function. The derivative represents the rate at which a function is changing at any given point. In our given function \( y^3 = x^5 \), differentiating both sides of the equation with respect to \( x \) involves applying differentiation rules such as the power rule and chain rule. Here鈥檚 how it goes step-by-step:
  • Directly differentiate the right-hand side of the equation using the power rule: since \( \frac{d}{dx}(x^5) = 5x^4 \).
  • For the left side, apply the chain rule to differentiate \( y^3 \). This results in \( 3y^2 \frac{dy}{dx} \).
  • You end up with: \( 3y^2 \frac{dy}{dx} = 5x^4 \).
By solving this equation for \( \frac{dy}{dx} \), you find the differential relationship between \(y\) and \(x\). Differentiation allows us to understand how a small change in \( x \) leads to a change in \( y \).
Calculus Techniques
Implicit differentiation and the chain rule are crucial calculus techniques that help solve complex functions. When faced with an equation where \( y \) is not explicitly solved for \( x \), we use implicit differentiation. Here is a breakdown:
  • Implicit Differentiation: This involves differentiating each term of an equation with respect to \( x \) and then solving for \( \frac{dy}{dx} \). It鈥檚 particularly useful when it's difficult or impossible to solve for \( y \) explicitly.
  • Chain Rule: As discussed, the chain rule is vital in dealing with nested functions. It simplifies the differentiation process by breaking it into manageable parts.
  • Power Rule: This basic rule of differentiation states that \( \frac{d}{dx}(x^n) = nx^{n-1} \). We used this to differentiate \( x^5 \), resulting in \( 5x^4 \).
  • Solving for \( \frac{dy}{dx} \): After applying differentiation rules, you solve for the derivative, \( \frac{dy}{dx} = \frac{5x^4}{3y^2} \). This formula indicates how \( y \) changes with \( x \) under the given relationship.
Mastering these techniques is essential for tackling more advanced problems in calculus and understanding how various functions behave.

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Most popular questions from this chapter

Cost and tolerance. A painting firm contracts to paint the exterior of a large water tank in the shape of a half-dome (a hemisphere). The radius of the tank is measured to be \(100 \mathrm{ft}\) with a tolerance of 卤6 in. \((\pm 0.5 \mathrm{ft}) .\) (The formula for the surface area of a hemisphere is \(A=2 \pi r^{2} ;\) use 3.14 as an approximation for \(\pi\).) Each can of paint costs \(\$ 30\) and covers \(300 \mathrm{ft}^{2}\) a) Calculate \(d A,\) the approximate difference in the surface area due to the tolerance. b) Assuming the painters cannot bring partial cans of paint to the job, how many extra cans should they bring to cover the extra area they may encounter? c) How much extra should the painters plan to spend on paint to account for the possible extra area?

Graph cach function. Then estimate any relative extrema. Where appropriate, round to three dreimal places. $$f(x)=x^{2}(x-2)^{3}$$

Assume the function \(f\) is differentiable over the interval \((-\infty, \infty)\) : that is, it is smooth and continuous for all real numbers \(x\) and has no corners or vertical tangents. Classify each of the following statements as cither true or false. If you choose false, explain why. The function \(f\) can have a point of inflection at a critical value.

Graph cach function. Then estimate any relative extrema. Where appropriate, round to three dreimal places. $$f(x)=x-\sqrt{x}$$

$$\text {Graph each function using a calculator, iPlot, or Graphicus.}$$ $$f(x)=\frac{x^{3}+2 x^{2}-3 x}{x^{2}-25}$$
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